Cotangent Periodicity Quiz: Cotangent Periodicity Odd Symmetry

1) For all θ where defined, cot(θ+π)=cotθ and cot(−θ)=−cotθ can both hold without contradiction.

True

False

One states period π; the other states odd symmetry. Both are standard identities and are compatible since f(θ+π)=f(θ) and f(−θ)=−f(θ) can hold simultaneously.

Explanation

One states period π; the other states odd symmetry. Both are standard identities and are compatible since f(θ+π)=f(θ) and f(−θ)=−f(θ) can hold simultaneously.

2) What is the fundamental period of y = cotθ?

π

2π

π/2

No period

cot(θ+π)=cos(θ+π)/sin(θ+π)=(-cosθ)/(-sinθ)=cosθ/sinθ=cotθ, so the least positive period is π.

Explanation

cot(θ+π)=cos(θ+π)/sin(θ+π)=(-cosθ)/(-sinθ)=cosθ/sinθ=cotθ, so the least positive period is π.

3) Select all true statements about cot symmetry and shifts.

Reflecting across the origin maps the graph to itself

Shifting by π maps the graph to itself

Reflecting across the y-axis maps the graph to itself

Cot(θ+π)=−cotθ

Cot(−θ)=cotθ

Odd functions have origin symmetry: (θ, y) → (−θ, −y). Period π means horizontal shift by π preserves the graph. The others are false.

Explanation

Odd functions have origin symmetry: (θ, y) → (−θ, −y). Period π means horizontal shift by π preserves the graph. The others are false.

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4) State all vertical asymptotes of cotθ on [0, 2π].

cotθ=cosθ/sinθ is undefined where sinθ=0, which happens at θ=kπ. On [0,2π], these are 0, π, 2π.

Explanation

cotθ=cosθ/sinθ is undefined where sinθ=0, which happens at θ=kπ. On [0,2π], these are 0, π, 2π.

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5) Select all angles in [−π, π] that map to the same cotangent value as θ = π/6.

θ = π/6

θ = 7π/6

θ = −5π/6

θ = −11π/6

θ = −5π/6 + π

Angles that differ by integer multiples of π have the same cot value. In [−π, π], π/6 and π/6±π→π/6 and −5π/6. 7π/6 is outside but equivalent modulo 2π; E simplifies to π/6.

Explanation

Angles that differ by integer multiples of π have the same cot value. In [−π, π], π/6 and π/6±π→π/6 and −5π/6. 7π/6 is outside but equivalent modulo 2π; E simplifies to π/6.

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7) Select all identities equivalent to the π-periodicity of cotθ.

Cot(θ+π)=cotθ

Cot(θ+2π)=cotθ

Cot(θ+π/2)=cotθ

Cot(θ+nπ)=cotθ for all integers n

Cot(θ+π)=−cotθ

The fundamental period is π, so integer multiples preserve value. π/2 does not. Saying cot(θ+π)=−cotθ is false; that would contradict periodicity.

Explanation

The fundamental period is π, so integer multiples preserve value. π/2 does not. Saying cot(θ+π)=−cotθ is false; that would contradict periodicity.

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8) Shifting the graph of cotθ right by π units reproduces the same graph.

True

False

Periodicity: cot(θ+π)=cotθ, so a horizontal shift by π yields the same values.

Explanation

Periodicity: cot(θ+π)=cotθ, so a horizontal shift by π yields the same values.

9) Because cot is odd, its graph passes through the origin.

True

False

cot0 is undefined (vertical asymptote at θ=0), so although the function is odd, it does not pass through the origin.

Explanation

cot0 is undefined (vertical asymptote at θ=0), so although the function is odd, it does not pass through the origin.

10) Which interval displays one full basic branch of y = cotθ?

(0, π)

(−π/2, π/2)

(0, 2π)

(π/2, 3π/2) including endpoints

Between consecutive asymptotes at kπ and (k+1)π, cotθ is continuous and strictly decreasing. (0, π) is one such branch; endpoints excluded since undefined.

Explanation

Between consecutive asymptotes at kπ and (k+1)π, cotθ is continuous and strictly decreasing. (0, π) is one such branch; endpoints excluded since undefined.

11) Complete the identity: cot(θ+π) = ____.

Using angle sum: cos(θ+π)=−cosθ and sin(θ+π)=−sinθ, so cot(θ+π)=cos(θ+π)/sin(θ+π)=(−cosθ)/(−sinθ)=cotθ.

Explanation

Using angle sum: cos(θ+π)=−cosθ and sin(θ+π)=−sinθ, so cot(θ+π)=cos(θ+π)/sin(θ+π)=(−cosθ)/(−sinθ)=cotθ.

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12) If cotβ = 1, which is also true?

Cot(−β) = −1

Cot(β+π) = −1

Cot(β+π/2) = 1

Cot(2β) = 1 always

Odd symmetry gives cot(−β)=−cotβ=−1. Periodicity means cot(β+π)=cotβ=1, not −1. No general identity fixes cot(2β) to 1.

Explanation

Odd symmetry gives cot(−β)=−cotβ=−1. Periodicity means cot(β+π)=cotβ=1, not −1. No general identity fixes cot(2β) to 1.

13) Which set best describes all angles where cotθ = 0?

θ = π/2 + kπ

θ = kπ

θ = π + 2kπ only

No solutions

cotθ=0⇔cosθ=0 with sinθ≠0. cosθ=0 at θ=π/2 + kπ for any integer k.

Explanation

cotθ=0⇔cosθ=0 with sinθ≠0. cosθ=0 at θ=π/2 + kπ for any integer k.

14) Select all statements that follow from cot being odd.

Cot(−θ)=−cotθ

The graph is symmetric about the origin

Cot(θ) = cot(−θ)

If cotα = k then cot(−α) = −k

Cot has period 2π

Oddness means f(−x)=−f(x). This gives origin symmetry and flips signs at opposite angles. Oddness does not imply period 2π; cot's period is π.

Explanation

Oddness means f(−x)=−f(x). This gives origin symmetry and flips signs at opposite angles. Oddness does not imply period 2π; cot's period is π.

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15) Cot(−θ) = −cotθ for all θ where both sides are defined.

True

False

cot(−θ)=cos(−θ)/sin(−θ)=(cosθ)/(−sinθ)=−(cosθ/sinθ)=−cotθ, which proves odd symmetry about the origin.

Explanation

cot(−θ)=cos(−θ)/sin(−θ)=(cosθ)/(−sinθ)=−(cosθ/sinθ)=−cotθ, which proves odd symmetry about the origin.

16) Complete: Because cot is odd, cot(π−θ) equals ____ times cot(θ−π).

cot(π−θ)=cot(−(θ−π))=−cot(θ−π). But θ−π is negative of (π−θ), so applying oddness twice yields cot(π−θ)=cot(θ−π). The multiplier is 1.

Explanation

cot(π−θ)=cot(−(θ−π))=−cot(θ−π). But θ−π is negative of (π−θ), so applying oddness twice yields cot(π−θ)=cot(θ−π). The multiplier is 1.

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17) Which transformation generates the next branch of y = cotθ from the branch on (0, π)?

Shift right by π

Shift left by π/2

Reflect across the y-axis

Shift up by π

Since cot is π-periodic, shifting horizontally by π reproduces the same shape to the right (or left).

Explanation

Since cot is π-periodic, shifting horizontally by π reproduces the same shape to the right (or left).

18) Using unit-circle coordinates (x,y)=(cosθ,sinθ), write cotθ in terms of x and y.

By definition, cotθ=cosθ/sinθ = x/y, provided y≠0.

Explanation

By definition, cotθ=cosθ/sinθ = x/y, provided y≠0.

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19) If cotγ = −√3, then cot(−γ) = √3.

True

False

Odd symmetry: cot(−γ)=−cotγ=−(−√3)=√3.

Explanation

Odd symmetry: cot(−γ)=−cotγ=−(−√3)=√3.

20) Given cotα = −2, what is cot(α+π)?

12/28

1/2

11/13

Undefined

Using period π: cot(α+π)=cotα=−2 (provided both angles avoid asymptotes).

Explanation

Using period π: cot(α+π)=cotα=−2 (provided both angles avoid asymptotes).